Theorem eqelssd 3246

Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
eqelssd.1	|- ( ph -> A C_ B )
eqelssd.2	|- ( ( ph /\ x e. B ) -> x e. A )

Assertion

Ref	Expression
eqelssd	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem eqelssd

Step	Hyp	Ref	Expression
1		eqelssd.1	. 2 |- ( ph -> A C_ B )
2		eqelssd.2	. . . 4 |- ( ( ph /\ x e. B ) -> x e. A )
3	2	ex 115	. . 3 |- ( ph -> ( x e. B -> x e. A ) )
4	3	ssrdv 3233	. 2 |- ( ph -> B C_ A )
5	1, 4	eqssd 3244	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   C_ wss 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3206  df-ss 3213

This theorem is referenced by:  fiuni  7176  ennnfonelemrn  13039  ennnfonelemdm  13040  unirnblps  15145  unirnbl  15146  dvidlemap  15414  dvidrelem  15415  dvidsslem  15416  dviaddf  15428  dvimulf  15429  dvcj  15432  dvrecap  15436